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Preface
This solution manual was prepared as an aid for instrctors who wil benefit by
having solutions available. In addition to providing detailed answers to most of the
problems in the book, this manual can help the instrctor determne which of the
problems are most appropriate for the class.
The vast majority of the problems have been solved with the help of available
computer software (SAS, S~Plus, Minitab). A few of^ the problems have been solved with
hand calculators. The reader should keep in mind that round-off errors can occur-
parcularly in those problems involving long chains of arthmetic calculations.
We would like to take this opportnity to acknowledge the contrbution of many
students, whose homework formd the basis for many of the solutions. In paricular, we
would like to thank Jorge Achcar, Sebastiao Amorim, W. K. Cheang, S. S. Cho, S. G.
Chow, Charles Fleming, Stu Janis, Richard Jones, Tim Kramer, Dennis Murphy, Rich
Raubertas, David Steinberg, T. J. Tien, Steve Verril, Paul Whitney and Mike Wincek.
Dianne Hall compiled most of the material needed to make this current solutions manual
consistent with the sixth edition of the book.
The solutions are numbered in the same manner as the exercises in the book.
Thus, for example, 9.6 refers to the 6th exercise of chapter 9.
We hope this manual is a useful aid for adopters of our Applied Multivariate
Statistical Analysis, 6th edition, text. The authors have taken a litte more active role in
the preparation of the current solutions manual. However, it is inevitable that an error or
two has slipped through so please bring remaining errors to our attention. Also,
comments and suggestions are always welcome.
Richard A. Johnson
Dean W. Wichern
1.2 a)
Xl =" 4.
51i = 4.
.ì
Chapter 1
X2 = 15.
522 = 3.56 S12 = 3.
Scatter Plot and Marginal Dot Plots
...^.^.^.^.^.^.
...^.
.. .
12~
..
I' )C
7..^. .. 5. (^)..
0 2 4 6 8 10 12 xl
b) SlZ is negative
c)
Xi =5.20 x2 = 12.48 sii = 3.09 S22 = 5. SI2 = -15.94 'i2 = -.
Large Xl occurs with small Xz and vice versa.
d)
x = 12. ( 3.
S -n - -15.
R =( 1 -.98)
1.5 a) There is negative correlation between X2 and X3 and negative correlation
between Xl and X3. The marginal distribution of Xi appears to be skewed to
the right. The marginal distribution of X2 seems reasonably symmetric.
The marginal distribution of X3 also appears to be skewed to the right.
Sêåttêr'Plotäl'(i'Marginal.DotPiØ_:i..~sxli.
1600
M)C (^). (^). (^)..^.
400
. (^)...^.^.^.^. 0
x
. '-' . .Scatiêr;Plötànd:Marginal.alÎ.'llÎi:lîtfjtì.I... ......^.^.^.^. 1600 ..
M )C 800. ..^.
400
....^.^.^. 0 50 100 150 200 250 300 xl
1.5 b)
(155.60J
x = 14.
R = ( ~ -.
1.6 a) Hi stograms
HIDDLE OF INTERVAL
HIDDLE OF INTERVAL
HIDIILE OF INTERVAL
J.
HIDDLE OF INTERVAL
- .. ... :.
s.
Sn = 273.
-.85)
1
. 1
Xi
Xs
HIDIILE OF NUMB£R. OF NUMBER OF INTERVAL 08SERVATIONS OBSERVATioNS cooJ.^2 **
5 *****^ 6.^ J^ U*
S ********^ 7. S *****
7 u***** a. s *****
11 ***********^ 9.^6 ******
5 un*^ 10. 4 ****
6 ******^ 11. 4 u**
12.. S Uu.*
X2 13.^4 ****
NUHBER OF 16. 1 *
OBSERVATIONS 17. 0
1 * (^) LS. 1 * J n* (^) 19. 0
a ******** Xl
2 n 1 * (^) I' I DOLE .OF INTE"RVAL
X
a.
NUHBER OF OBSERVA T I'ONS 1 *
:s *****
19 ******************* 9 ********* .3 U* S u***
X
NUMBER OF. OBSERVA T IONS 1 J *$********* 15 *************** a ******** 5 ui 1 *
HIDDLE OF INTERVAL
J.
s.
NUMBER OF OBSERVATIONS 7 ******* 25 ************************* 9 ********* 1 *
NUltEiER OF OI4SERVATIOllS J *** 4 ****
7 ******* B ******** 5 n*** 2 ** 2 u 1 * o o 2 ** 1 *
X
1.8 Using (1-12) d(P,Q) = 1(-1-1 )2+(_1_0)2; = /5 = 2.
Using (1-20) d(P.Q)' /~H-1 )'+2(l)(-1-1 )(-1-0) '2t(-~0);' =j~~ = 1.38S
Using (1-20) the locus of points a c~nstant squared distance 1 from Q = (1,0)
is given by the expression t(xi-n2+ ~ (x1-1 )x2 + 2t x~ = 1. To sketch the
locus of points defined by this equation, we first obtain the coordinates of
some points sati sfyi ng the equation:
(-1,1.5), (0,-1.5), (0,3), (1,-2.6), (1,2.6), (2,-3), (2,1.5), (3,-1.5)
The resulting ellipse is:
X
1.9 a)^ sl1^ =^ 20.48^ s 22 = 6. 19^
s 12 = 9.
X
. .
..
xi
. . . -"
1.10 a) This equation is of the fonn (1-19) with aii = 1, a12 = ~. and aZ2 = 4.
Therefore this is a distance for correlated variables if it is non-negative
for all values of xl' xz' But this follows^ easily if we write
- 1 1 15 2. xl + 4xZ + x1x2 = (xl + r'2) + T x2 ,?o.
b) In order for this expression to be a distance it has to be non-negative for
- :¿ all values xl' xz' Since, for (xl ,x2) = (0,1) we have xl-2xZ = -Z, we
conclude that this is not a validdistan~e function.
d(P,Q) = 14(X,-Yi)4 + Z(-l )(x1-Yl )(x2-YZ) + (x2-Y2):¿'
= 14(Y1-xi):¿ + 2(-i)(yi-x,)(yZ-x2) + (xz-Yz):¿' = d(Q,P)
Next, 4(x,-yi)2. - 2(xi-y,)(x2-y2) + (x2-YZ): =
=,(x1-Yfx2+Y2):1 + 3(Xi-Yi):1,?0 so d(P,Q) ~O.
The s€cond term is zero in this last ex.pr.essi'on only if xl = Y1 and
then the first is zero^ only if x.2 = YZ.
1.14 a)
360.+ )(
320.+
280.+
240.+
200.+
.
. .
.. . .
. ....
.. .. . .. I: I:
. ... .. .
160.+ (^) +______+_____+-------------+------~.. )(
- 1:5:5. 180. 20:5.' 230. 2:5:5.
Strong positive correlation. No obvious "unusual" observations.
b) Mul tipl e-scl eros; s group.
x =
- 07
116.91 61 .78^ -20.10^ 61. 1 3^ -27. 65
812.72 -218.35^ 865.32^ 90.
Sn =^ 3 as. 94^ 221 '. 93^ 286.
(synetric) 337.
Non multiple-sclerosis group.
s =n
1.15 4.61^ ..92^.^
.61 .11^ .12^ .39^ -.
.57 .09 .34^.
Sn =^ .11 .21.
;. (synetric).
1 .551^ .362^ .386^ .537^. 077 '
1 .187^ .455^ .535^ -.
1 .346 .496^.
R =^1 .704.
1 -. 01 0
(syretric) 1
The largest correlation is between appetite and amount of food eaten.
Both activity and appetite have moderate positive correlations with
symptoms. A1 so, appetite and activity have a moderate positive
correl a tion.
There are signficant positive correlations among al variable. The lowest correlation is
. 0.4420 between Dominant humeru and Ulna, and the highest corr.eation is 0.89365 bewteen
Dominant hemero and Hemeru.
There are large positive correlations among all variables. Paricularly large
correlations occur between running events that are "similar", for example,
1.19 (a)
ULNA ILULNA^ tlUME~US^ LHUI.ERUS^
RADIUS o _R A 0 IUS
I
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1.19 (b)
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Outlier
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tfe'
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Outlier
X
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Outlier Q
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(a) There are two outliers in the upper right and lower right corners of the plot.
(b) Only the points in the gasoline group are highlighted. The observation in the upper
right is the outlier. As indiCated in the plot, there is an orientation to classify into two
groups.
1.22 possible outliers are indicated.
G
Outlier
X
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